© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Efficient Fault-Tolerant Quantum Computing Martin Suchara AT&T Labs September 8, 2015

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Applications of Quantum Computing  Shor’s factoring algorithm Find prime factors of integer N Quantum algorithm runs in polynomial time Can be used to break public-key cryptography (RSA)  How can we build a scalable reliable quantum computer? 2

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Some Candidate Hardware Technologies  Superconducting qubits Josephson Junctions between superconducting electrodes 3  Ion traps Ions trapped in electromagnetic field, gates performed by applying lasers  Neutral atoms Ultracold atoms trapped by light waves in an optical lattice 3

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Quantum Error Correction (QEC)  Much more challenging than classically Analog nature of quantum operations New kinds of errors: partial bit flips, phase flips, small shifts  Need error correction for reliable information storage and computation with unreliable technologies  State can even leak out of the code space 4

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Overview I.How much resources needed by quantum computing?  ICCD 2013 Conference, Quantum Information and Computation 2014 II.Improved error decoding – MLE algorithms III.New types of errors – correcting qubit leakage IV.Future directions 5

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Shor’s Factoring Algorithm – How Long to Break a Key?  Algorithm needs approximately 1.68 x 10 8 Toffoli gates and 6,144 logical qubits (Jones et al., 2012)  Factor a 1024-bit number 6

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Properties of Future Technologies (Estimates in IARPA QCS Project)  The speed and reliability of gates varies  Much higher errors than classical computers 7

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. QuRE: The Quantum Resource Estimator Toolbox Algorithm Specs Technology Specs Analysis of Error Correction  # of logical qubits  # of logical gates  Circuit parallelism  Gate times and fidelities  Memory error rates  Estimate cost of each logical operation as a function of error correction “strength” Automated Resource Estimate  Find code distance / concatenations needed for successful error correction  Estimate number of physical qubits, running time, physical gate and instruction count, etc. 8

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Concatenated Error Correction Codes – Steane [[7,1,3]] Code  Multiple qubits encode a single logical qubit  Most operations transversal:  Non- transversal T gate: 9

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Concatenated Error Correction Codes – Tiled Qubit Layout  Each logical qubit is stored in a separate tile  Tiles are hierarchical  Supported operations: Error correct a tile Apply fault-tolerant logical operation  Tiles must contain enough data and ancilla qubits 10

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Optimized Layout in Each Tile (Svore et al., 2006) “empty” qubit data qubit ancilla qubit SWAP CNOT verification qubit 11

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Topological Quantum Error Correction – The Surface Code  Physical qubits on links in the lattice  Measuring the shown “check” operators yields error syndromes 12

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Topological Quantum Error Correction – Example of Errors and Syndromes  Guess a likely error consistent with observed syndromes  Error correction performed continuously 13

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Topological Quantum Error Correction – Tiles Represent Logical Qubits  Each logical qubit represented by a pair of holes  CNOT gates performed by moving holes around each other additional space for CNOTs and magic state distillation 14

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Numerical Results – Shor’s Factoring e = 1 x t = 19,000 ns e = 1 x t = 25 ns e = 1 x t = 32,000 ns 15

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© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Fault tolerance increases number of qubits by 2-4 orders of magnitude and number of gates by 9-10 orders of magnitude! T gates are the most expensive 17

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Overview I.How much resources needed by quantum computing? II.Improved error decoding – MLE algorithms  Physical Review A, 2014 III.New types of errors – correcting qubit leakage IV.Future directions 18

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Goal – Construct a Maximum Likelihood Decoder for the Surface Code  Error decoders use a heuristic - minimum weight matching - to guess a likely error Problem 1: ignores degeneracy, existence of multiple error chains between a specific syndrome pair Problem 2: ignores correlations of X and Z errors in some error models  Goal: construct a decoder that finds the most likely error given the observed syndrome 19

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Degeneracy  A possible syndrome measurement 20

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Degeneracy  Two ways of matching with equal cost, matching algorithm picks one at random X X X X X X X X 21

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Degeneracy  The green error less likely than one of the red ones, which are equivalent X X X X X X X X 22

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Our Maximum Likelihood Decoder (MLD)  Maximum likelihood decoder finds the most likely error given the observed syndrome Must consider all error chains Works by formulating problem as a matchgate quantum circuit that can be simulated efficiently Exact solution in time O(n 2 ) where n is the number of code qubits Only works for X error noise: 23

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Threshold of the ML Decoder, X-Noise Model ~ 10.9% threshold agrees with phase transition of a spin model 24

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Correlations  Example of a syndrome measurement  Assume 25

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Correlations  Pair up syndromes and correct errors Z Z Z X X X 26

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Inefficiency of Minimum Weight Matching (MWM) – Correlations  Actual error (in red) much more likely cause of syndrome! Logical error occurs. Z Z Z X X X Y Y Y 27

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Approximate Solution for More General Noise Models  Approximate algorithm that uses matrix product states (MPS) Approximate solution in time O(n ) where controls the approximation precision Close to optimum for small Works for the depolarizing noise model: Key step contracts a tensor network on the two-dimensional grid of the code 28

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Threshold of the ML Decoder, X-Noise Model 29 Threshold above 18%

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Exact and Approximate ML Decoding, X-Noise Model Evidence that MPS is nearly optimal 30

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Approximate ML Decoding with the MPS Algorithm, Depolarizing Noise 31

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. 100 fold or better improvement of logical error rates in depolarizing noise models! Considering correlations much more important than degeneracy Open question: maximum likelihood decoding with noisy syndrome extraction 32

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Overview I.How much resources needed by quantum computing? II.Improved error decoding – MLE algorithms III.New types of errors – correcting qubit leakage  Quantum Information and Computation 2015, ISIT 2015 Conference IV.Future directions 33

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. What is Qubit Leakage?  Physical qubits are not ideal two-level systems and may leak out of the computational space  Our work: simple model of leakage and comparisons of leakage reduction strategies  With standard error correction techniques leaked qubits accumulate and spread errors Bit flip Leakage 34

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Leakage in the Literature  Analysis of leakage reduction units based on quantum teleportation, threshold theorem for concatenated codes (Aliferis, Terhal 2005)  Model of leakage for repetition code that labels leaked qubits (Fowler 2013) 35

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Simple Model of Error Correction with the Toric Code  Label-based model: each qubit is in state I, X, Y, Z, or L Data qubitAncilla Syndrome extractions: 36

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Monte Carlo Simulations: Measure Syndromes and Decode Errors  Use minimum weight matching decoder and correct errors between matched syndrome pairs  Measure syndromes, build a 3-D syndrome history (a 2-D slice shown) 37

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Leakage Model of Gates 38

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Leakage Reduction Circuits 1.Full-LRU: (resource heavy) 3. Quick circuit: (swap data and ancilla) 2. Partial-LRU: (fewer gates) 39

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. The Standard and Heralded Leakage (HL) Decoders  Standard Decoder only relies on syndrome history to decode errors  HL Decoder uses leakage detection when qubits are measured  Partial information about leakage locations  Error decoder must be modified 40

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Standard Decoder for the Toric Code  Corrects error chains between pairs of matched syndromes  Need to adjust edge weights for each leakage suppressing circuit (Full-LRU, Partial-LRU, Quick circuit)  Decoding graphs for X and Z errors built up using this unit cell (Fowler 2011) 41

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. HL Decoder – Quick Circuit 42

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Threshold Comparison  More complicated circuits have lower threshold  HL decoder helps boost the threshold 43

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Decoding Failure Rates  Full-LRU performs well at low error rates 44

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. 45 Leakage reduction is necessary A model of leakage and systematic exploration of parameter space A simple leakage reduction circuit that only adds a single CNOT gate and new decoders are effective

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Overview I.How much resources needed by quantum computing? II.Improved error decoding – MLE algorithms III.New types of errors – correcting qubit leakage IV.Future directions 46

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Improving the Efficiency of Computation with the Surface Code  Implementation of CNOT gates by braiding of smooth and rough holes in the surface A few known transformation rules can make the volume of the code smaller Complete set of rules and their optimal use? 47

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. Decreasing the Overhead of T Gates  T gates are very expensive  New codes with transversal T gates No go theorem of Bravyi and Konig doesn’t apply to subsystem codes Are there 2-D subsystem codes with transversal T gates and good enough error suppression? 48

© 2015 AT&T Intellectual Property. All rights reserved. AT&T, the AT&T logo and all other AT&T marks contained herein are trademarks of AT&T Intellectual Property and/or AT&T affiliated companies. The information contained herein is not an offer, commitment, representation or warranty by AT&T and is subject to change. 49 Thank You!